Subgroups — #30770543

Order 17152.

398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562459	C₂	1	4	yes	maximal, complemented
2	#22559824	C₂	1	2	yes	maximal, complemented
3	C₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₇₂	—	2	8	no	—
5	#16863773	—	2	4	no	—
6	#16863773	C₂²	1	4	yes	—
7	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
8	C₈²×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×D₅₃₆	—	2	8	no	—
10	C₂×D₅₃₆	—	4	8	no	—
11	#14524040	—	2	4	no	—
12	D₁₀₇₂	—	4	16	no	—
13	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
14	C₄×C₈×C₆₇	D₄	1	2	yes	—
15	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
16	C₂×D₂₆₈	—	4	4	no	—
17	C₂×D₂₆₈	—	4	8	no	—
18	D₅₃₆	—	4	16	no	—
19	D₅₃₆	—	8	16	no	—
20	C₂×C₈×C₆₇	D₈	1	4	yes	—
21	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
22	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
23	C₁₆×C₆₇	D₈	1	8	yes	complemented
24	C₂²×D₆₇	—	8	4	no	—
25	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
26	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
27	D₂₆₈	—	8	8	no	—
28	D₂₆₈	—	8	16	no	—
29	C₈×C₆₇	D₁₆	1	8	yes	complemented
30	C₈×C₆₇	C₂×D₈	1	4	yes	—
31	C₂²×C₆₇	#493	1	1	yes	characteristic
32	C₂×D₆₇	—	16	4	no	—
33	C₂×D₆₇	—	16	8	no	—
34	C₄×C₆₇	#493	1	2	yes	—
35	C₄×C₆₇	C₂×D₁₆	1	4	yes	—
36	#12515	—	67	1	no	maximal, Sylow
37	C₂×C₆₇	#1712	1	1	yes	characteristic
38	C₂×C₆₇	#2246	1	2	yes	—
39	D₆₇	—	32	4	no	—
40	#2246	—	67	4	no	—
41	#1712	—	67	2	no	—
42	C₈×C₁₆	D₆₇	1	1	yes	characteristic, complemented
43	C₆₇	#12515	1	1	yes	characteristic, complemented, Sylow
44	C₂×D₁₆	—	134	8	no	—
45	#493	—	67	4	no	—
46	#493	—	134	4	no	—
47	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
48	C₈²	C₂×D₆₇	1	1	yes	characteristic
49	C₂×D₈	—	268	8	no	—
50	C₂×D₈	—	134	8	no	—
51	#127	—	134	4	no	—
52	D₁₆	—	268	16	no	—
53	C₂×C₁₆	D₂₆₈	1	4	yes	—
54	C₄×C₈	D₂₆₈	1	2	yes	—
55	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
56	C₂×D₄	—	268	4	no	—
57	C₂×D₄	—	268	8	no	—
58	D₈	—	268	16	no	—
59	D₈	—	536	16	no	—
60	C₂×C₈	D₅₃₆	1	4	yes	—
61	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
62	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
63	C₁₆	D₅₃₆	1	8	yes	complemented
64	C₂³	—	536	4	no	—
65	D₄	—	536	8	no	—
66	D₄	—	536	16	no	—
67	C₂×C₄	#14524040	1	1	yes	characteristic
68	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
69	C₈	D₁₀₇₂	1	8	yes	complemented
70	C₈	C₂×D₅₃₆	1	4	yes	—
71	C₂²	—	1072	4	no	—
72	C₂²	—	1072	8	no	—
73	C₂²	#16863773	1	1	yes	characteristic
74	C₄	#16863773	1	2	yes	—
75	C₄	C₂×D₁₀₇₂	1	4	yes	—
76	C₂	—	2144	4	no	—
77	C₂	#22559824	1	1	yes	characteristic
78	C₂	#22562459	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863773
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₈×C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	#493
hypercenter	C₈×C₁₆	D₆₇
hypocenter	C₆₇	#12515
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22562459	C₂	semidirect
2. #22559824	C₂	semidirect
3. C₈×C₁₆×C₆₇	C₂	semidirect
23. C₁₆×C₆₇	D₈	semidirect
29. C₈×C₆₇	D₁₆	semidirect
42. C₈×C₁₆	D₆₇	semidirect
43. C₆₇	#12515	semidirect
63. C₁₆	D₅₃₆	semidirect
69. C₈	D₁₀₇₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22562459	2
#22559824	2
C₈×C₁₆×C₆₇	2
#12515	67